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  1. ; ; ; (, Compositio Mathematica)
    null (Ed.)
    We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $$X \longrightarrow B$$ with singular fibre over $$b_0\in B$$ yields a family $$\mathscr {M}(X/B,\beta ) \longrightarrow B$$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $$b_0$$ in terms of rigid tropical maps to the tropicalization of $X/B$ . This generalizes one aspect of known results in the case that the fibre $$X_{b_0}$$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples. 
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  2. ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract In this paper, we study {\mathbb{A}^{1}} -connected varieties from log geometry point of view, and prove a criterion for {\mathbb{A}^{1}} -connectedness. As applications, we provide many interesting examples of {\mathbb{A}^{1}} -connected varieties in the case of complements of ample divisors, and the case of homogeneous spaces. We also obtain a logarithmic version of Hartshorne conjecture characterizing projective spaces and affine spaces. 
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  3. ; (, Selecta Mathematica)
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  4. ; (, Annales de l'Institut Fourier)
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  5. ; (, Journal of Algebraic Geometry)
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